Mathematics

UNIVERSITY OF MILANO-BICOCCA
A.Y. 2007/2008
MATHEMATICS
Code
517011
Title of the course
Credits
5
Field
INF/01
Year of course
1
Semester
n.d.
Assesment
method
Scritto; Voto finale
Lecturer
Programme
Aim: the aim of this corse is to introduce the basic concepts related to the architecture of computer
systems and to supply the student the basic competences required to define the algorithmic solution
to simple problems, coding the solution in Java language.
Prerequisites: no required prerequisites.
Main Topics:
Architecture of computer systems:
* Computer archtecture basics and information coding.
* Operating systems basics.
* Computer networks basics.
Imperative programming in Java:
* Programming language hierarchy, compilers and interpreters·
* The Java Virtual Machine.·
* Algorithms and programs.·
* Primitive data types.·
* Control flow structures: choices and loops.·
* Arrays of primitive data types.·
* Methods: definition and invocation.
Bibliographic references: bibliography will be given at the beginning of the course.
Tests: rules will be explained at the beginning of the course.
Code
517093
Title of the course
Credits
5
Field
MAT/08
Year of course
1
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Russo Alessandro
Programme
Aim: to present some elementary techniques of numerical analysis in order to promote the use of
computers for mathematical investigations. The software Matlab will be used.
Prerequisites: basics of linear algebra and calculus.
Pag. 1/22
Main Topics:
* Introduction to Matlab.
* Floating Point Arithmetic.
* Numerical Linear Algebra: Gaussian Elimination, PA=LU and Cholesky Decomposition.
* Numerical Integration.
* Zero-finding Methods.
Bibliographic references: the course notes will be made available.
Tests: written, oral tests, and reports on the laboratory activities.
Code
517091
Title of the course
Algebra I
Credits
5
Field
MAT/02
Year of course
1
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Weigel Thomas stefan
Programme
Main Topics:
* Sets, relations, operations.
* Groups (cyclic groups, fundamental theorems on morphisms, group actions, the symmetric
group, subgroups and cosets).
* Langrange's theorem and the orbit formula.
* Rings and modules, ideals and annihilators; algebras.
* The chinese reminder theorem.
* Commutative rings and fields. Noetherian domains, principal ideal domains, euclidean domains.
* The decomposition in products of prime numbers.
* Structure theorems for modules over PID. Polynomial rings.
* Torsion modules.
* Canonical forms of matrices, the Jordan form.
Bibliographic references:
S. Bosch, Algebra, Springer, 2003.
N. Jacobson, Basic Algebra I, Freeman & Co, 1985.
Exams: written and oral exams.
Code
517092
Title of the course
Geometry and Topology I
Credits
10
Field
MAT/03
Year of course
1
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Ferrario Davide luigi
Programme
Aim: the aim is to give an introductory course on geometry and topology, focusing on the topology of
the real line and the corresponding geometry of euclidean, affine and projective spaces.
Prerequisites: --.
Main Topics:
Pag. 2/22
* Metric spaces and continuity: topology of metric spaces.
* Properties of open subsets.
* Closed subsets of a metric space. Closure.
* Limit points. Topological spaces.
* Basis for a topology.
* Induced topology. Continuous maps and homeomorphisms.
* Product topology. Equivalence relations.
* Hausdorff spaces. Compactness.
* Compactness for metric and Euclidean spaces.
* Connected and path-connected spaces.
* Examples of topological groups and transformation groups.
* Geometry of affine spaces.
* Affine subspaces, Grassmann formula.
* Affine structure of a vector space.
* Affine maps. Incidence structure and parallelism.
* Affine Euclidean spaces.
* Orthogonal group.
* Classical transformation groups and finite subgroups.
* Projective spaces.
* Projectivity and projective frames, homogeneous coordinates.
* Projective closure of an affine space, points at infinity, affine charts on a projective space.
* Projective closure of an affine curve.
* Quadrics and conics.
Bibliographic references:
E. Sernesi, Geometria, vol.I-II, Bollati-Boringhieri, 1989, 1994.
H.S.M. Coxeter, Introduction to geometry, John Wiley and Sons, 1961, 1969, 1989.
M. Nacinovich, Elementi di geometria analitica, Serie di matematica e fisica, Napoli Liguori Editore,
1996.
Tests: written and oral test.
Code
517090
Title of the course
LINEAR ALGEBRA AND GEOMETRY
Credits
10
Field
MAT/02, MAT/03
Year of course
1
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Dalla volta Francesca
Programme
Aim: this is a first course in linear algebra and geometry, which gives the basis for deeper arguments
fundamental in Mathematics and Physics.
Prerequisites: a good knowledge of the mathematics studied in higher school.
* Vector spaces.
* Duality.
* Linear maps and systems of linear equations.
* Affine Geometry.
* Matrix calculus.
* The determinant.
* Eigenvalues, eigenvectors, diagonalization.
* Similarity, minimal polynomial.
* Scalar and Hermitian products.
* Euclidean Geometry.
* Self-adjoint, orthogonal, Hermitian, unitary, normal operators.
* Spectral theorem.
* Canonical forms.
Pag. 3/22
* Simultaneous diagonalization, uncertainty principle.
* Esponential of a matrix, square root of a positive definite matrix.
* Conics.
Bibliographic references:
Main references:
S. Lang, Algebra Lineare, ed. italiana, Boringhieri, 1970.
Other references:
M. Abate, Geometria, McGraw Hill, 2002.
F. Dalla Volta, M. Rigoli, Elementi di Matematica discreta e di Algebra lineare, Pearson Education,
2007
Tests: a written test based on exercises and a test on the theory.
Code
517010
Title of the course
MATHEMATICAL ANALYSIS 1
Credits
10
Field
MAT/05
Year of course
1
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Soardi Paolo maurizio
Programme
The course aims at providing a fairly thorough theoretical exposition of sequences, series,
differentiation and Riemann integration in one variable. It also aims at providing the background for
the extension to several variables.
Students must pass a written test before being admitted to the oral examination. Both exams can be
repeated in case of failure.
Code
517085
Title of the course
Physics 1 (I part)
Credits
5
Field
FIS/01
Year of course
1
Code
517071
Title of the course
Physics 1 (II part)
Credits
10
Field
FIS/01
Year of course
1
- MUTUATO
- MUTUATO
Pag. 4/22
Code
A5170053
Title of the course
Algebra II
Credits
5
Field
MAT/02
Year of course
2
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Di martino Lino giuseppe
Programme
Aim: The course will illustrate: a) further basic topics in the theory of rings and fields; b) the theory of
finitely generated modules over principal domains, with applications to abelian groups and linear
algebra.
Prerequisites: the content of the course Algebra - I mod.-.
Main Topics:
* Commutative rings: Noetherian domains. Polynomial domains; Hilbert's Nullstellensatz.
Localisation, local rings.
* Ring and field extensions: Algebraic and transcendental extensions; splitting field of a
polynomial; finite fields.
* Modules over rings and linear algebra. Free modules: bases, rank, universal property. Torsion.
Modules over principal domains: finitely generated modules; matrix equivalence and reduction to
normal form;. Structure theorem for finitely generated modules. Torsion modules and primary
decomposition. Invariant factors, elementary divisors. Applications to abelian groups and matrices:
Structure theorem for finitely generated abelian groups. Canonical forms of matrices: companion
matrix; rational canonical form; Jordan canonical form.
Bibliographic references:
N. Jacobson, Basic Algebra I, Freeman & Co, 1985.
B. Hartley, T. Hawkes, Rings, modules and linear algebra, Chapman & Hall, 1970.
S. Bosch, Algebra, Springer-Verlag, 2003.
Tests: written and oral test.
Code
517074
Title of the course
ALGORITHMS AND PROGRAMMING
Credits
5
Field
INF/01
Year of course
2
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Programme
Aim: to learn the basics of object-oriented programming with Java.
Main Topics:
not available at the publication date.
Bibliographic references: bibliography will be given at the beginning of the course.
Tests: written and oral test.
Pag. 5/22
Code
A5170057
Title of the course
An Introduction to Numerical Analysis
Credits
10
Field
MAT/08
Year of course
2
Semester
n.d.
Assesment
method
Orale; Voto finale
Lecturer
Bozzini Maria tugomira
Programme
•
Floating-point arithmetic, errors, numerical models and algorithms
•
Linear systems. Condition number. Direct methods, iteration methods, the coniugate
gradient method
•
The matrix eigenvalue problem. The power method and its variants
•
Approximation of data and functions. Polynomial interpolation. Piecewise polynomial
interpolation. The best approximation methods. Orthogonal polynomials.
•
Numerical Integration. Newton-Cotes formulas. Gaussian quadrature. Composite rules.
Adaptive quadrature.
•
Finite differences. Linear difference equations.
•
Numerical methods for ordinary differential equations. Runge-Kutta methods. Multistep
methods. Stability of numerical methods. Stiff problems.
Code
517018
Title of the course
CLASSICAL MECHANICS
Credits
10
Field
MAT/07
Year of course
2
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Magri Franco
Programme
Aim: the basic ideas of Mechanics, from Newton to Lagrange, Hamilton and Jacobi are introduced.
Examples are borrowed from Celestial Mechanics and from the mechanics of rigid bodies.
Prerequisites: --.
Main Topics:
* Introduction to the history of Mechanics. Nicolò di Cusa and the principle of virtual works. Kepler
and the motion of planets. The Mechanics of Newton and Euler. The geometrization of Mechanics by
Lagrange. The great synthesis by Hamilton and Jacobi. The crisis of foundations.
* The Mechanics of Lagrange and Hamilton. Systems and constraints. The configuration space
and its tangent bundle. The action of Maupertuis and the central equation. The equations of
Lagrange and Hamilton. Variational formulation.
* The geometrization of Mechanics. The symplectic structure. Symmetries and conservation laws.
Canonical transformation and the Hamilton-Jacobi equation.
* Examples. Motion of planets. Integrable tops. Coupled oscillators. Examples of integrable and
separable systems.
* Elements of the qualitative study of the motion.
Bibliographic references:
B. Silvestre-Brac, C. Gignoux, Mécanique. De la formulation lagrangienne au chaos hamiltonien,
EDP Sciences.
D.T. Greenwood, Advanced Dynamics, Cambridge university press, 2003.
N.M.J. Woodhouse, Introduction to Analytical Dynamics, Oxford Clarendon press, 1987.
Pag. 6/22
Tests: written and oral test.
Code
A5170058
Title of the course
Computational Geometry
Credits
5
Field
MAT/08
Year of course
2
Semester
n.d.
Assesment
method
Orale; Voto finale
Lecturer
Rossini Milvia francesca
Programme
Aim: the aim of the course is to learn how to design curves satisfying to those fundamental
properties essentials for the realization of industrial objects.
Prerequisites: --.
Main Topics:
* Introduction. Parametric curves, control polygon, convex hull.
* Bezier Curves: properties. The de Casteljau's algorithm, subdivision, rational Bezier curves.
* Spline curves. Definitions. Properties. De Boor's algorithm, multiple knots, multiple control points.
Refinement. The Oslo algorithm.
* Parametric and geometric continuity.
Bibliographic references: J. Hoschek, D. Lasser, Fundamental of Computer aided geometric design,
Wellesley, MA: A. K. Peters Ltd., 1993.
G. Farin, Curves and Surfaces for CAGD, Academic Press, 1993.
Tests: oral test.
Code
A5170054
Title of the course
Differential Equations and Mathematical Models
Credits
5
Field
MAT/07
Year of course
2
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Programme
Aim: the student becomes familiar with the idea of mathematical model of a natural phenomenon.
Furthermore, he/she is provided with some techniques for the qualitative study of the system of
differential equation that translates the phenomenon in a mathematical problem.
Prerequisites: elementary differential calculus
Main Topics:
* Discrete time models. Iterate maps and their representation. Fixed points and cobweb. Cascade
bifurcations. Logistic Map.
* One-dimensional systems. Flow in the line. Fixed points and stability. Some examples.
Bifurcations.
* Two-dimensional systems. Flows in the plane and their phase portrait. Linear systems.
Conservative systems. Some examples. An outline of periodic orbits and limit cycles.
Bibliographic references: S. Strogatz, Nonlinear Dynamics and Chaos, Addison Wesley, 1994.
Pag. 7/22
Tests: written and oral test.
Code
517020
Title of the course
DIFFERENTIAL GEOMETRY
Credits
5
Field
MAT/03
Year of course
2
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Ghigi Alessandro callisto
Programme
Aim: the course is an introduction to the geometry of curves and surfaces in Euclidean space.
Prerequisites: a good knowledge of the foundations of calculus, linear algebra and topology.
Main Topics:
* Smooth functions of a real variable. Plane and space curves. Frenet formulas, curvature and
torsion.
* Implicit function theorem. Regular surfaces. Differential calculus on a surface. Orientation.
* Local geometry of surfaces. First and second fundamental form. Normal curvature, principal
curvatures and Gaussian curvature.
* Intrinsic geometry of surfaces.Covariant derivative. Theorema Egregium. Parallel transport.
Geodesics. Geodesic curvature.
* Topology of compact surfaces (sketch). Triangulations. Gauss-Bonnet theorem
Bibliographic references: M.P. Do Carmo, Differential geometry of curves and surfaces, PrenticeHall, Inc., 1976.
E. Sernesi, Geometria 2, Bollati Boringhieri, 1994.
M. Abate, F. Tovena, Curve e superfici, Springer, 2006.
Tests: written and oral test.
Code
517017
Title of the course
MATHEMATICAL ANALYSIS 2
Credits
10
Field
MAT/05
Year of course
2
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Colzani Leonardo
Programme
Program: 1 - Differential calculus in several variables.
2 - Integral calculus in several variables.
3 - Curves, surfaces, differential forms.
4 - Sequences and series of functions.
5 - Differential equations.
Text: Enrico Giusti "Analisi Matematica 2", Editore Bollati Boringhieri.
Pag. 8/22
Code
A5170055
Title of the course
Measure theory
Credits
5
Field
MAT/05
Year of course
2
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Terracini Susanna
Programme
Aim: this is a complement to the course of Analisi Matematica 2 and is devoted to deepen some
aspects of the theory of integrals.
Prerequisites: Analisi Matematica 1 and 2.
Main Topics:
* Lebesgue measure.
* Lebesgue integral. Comparison with Riemann.
* Product measure and Fubini-Tonelli's theorem.
* Integration of differential forms.
* Stokes theorem.
Bibliographic references: bibliography will be given at the beginning of the course.
Tests: written and oral test.
Code
A5170056
Title of the course
PROBABILITY I
Credits
10
Field
MAT/06
Year of course
2
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Tessitore Gianmario
Programme
Introduction: The purpose is to give the basic ideas and the rigorous mathematical tools needed in
the study of random phenomena,
Required knowledge: calculus in one and more variables. Sequences and series. Lebesgue
integration.
Programma:
•
Axioms of Probability
•
Conditional probability and independence.
•
Probabilities on countable spaces
•
Random variables on countable spaces
•
Classical laws on countable spaces (Bernoulli, binomial, geometric, poissonian).
•
Construction of probability measures
•
Probability measures on the real line R.
•
Random variables and their laws
•
Absolutely continuous random variables and densities.
•
Uniform, gaussian, exponential, gamma laws.
•
Integration with respect to probability measures (mean, variance moments).
•
Independent random variables and product measures
•
Probabilities on Rn
•
Characteristic functions
•
Sums of independent random variables
Pag. 9/22
•
Gaussian random variables
•
Convergence of random variables: in probability, almost sure, weak
•
Law of large numbers
•
Central limit theorem
Suggested textbooks: J. Jacod, P. Protter, Probability Essentials, Springer, 2000.
P. Baldi, Calcolo delle Probabilità e Statistica, McGraw-Hill libri Italia, 1998.
Exam: written and oral (in a short time delay)
Code
A5170069
Title of the course
Advanced Algebra I
Credits
5
Field
MAT/02
Year of course
3
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Di martino Lino giuseppe
Programme
Aim: the course offers an introduction to the theory of Lie algebras, focusing on the classification of
finite-dimensional semisimple Lie algebras over an algebraically closed field of characteristic zero.
Lie algebras have important applications in several areas of mathematics and physics. In the
classical case, sufficiently deep and satisfactory results can be obtained with very limited
prerequisites.
Prerequisites: Linear algebra; basic algebra courses.
Main Topics:
I: Basic concepts.
* Definitions and examples.
* Ideals, homomorphisms, automorphisms.
* Nilpotent and solvable Lie algebras. Engel's theorem.
II: Semisimple Lie algebras.
* Theorems of Lie and Cartan.
* The Killing form.
* Representations: complete reducibility (Weyl's theorem).
* Representations of sl(2).
* Cartan decomposition (root spaces).
III: Root systems.
* Axioms.
* Simple roots and Weyl group.
* Classification of root systems.
* Construction of root systems.
IV: Complements.
* Isomorphism and conjugacy theorems (A sketch).
* The simple algebras (the classical algebras, the algebra G2).
Bibliographic references:
J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer Verlag, 1972.
N. Jacobson, Lie Algebras, Wiley Interscience, 1962.
Tests: written and oral test.
Pag. 10/22
Code
A5170045
Title of the course
Algorithms and Data Structures (Soft Computing) - MUTUATO
Credits
6
Field
INF/01
Year of course
3
Code
A5170052
Title of the course
BiologiaComputational methods in biology - MUTUATO
Credits
6
Field
INF/01
Year of course
3
Code
A5170046
Title of the course
Cybernetics (Elements of System Theory) - MUTUATO
Credits
6
Field
INF/01
Year of course
3
Code
A5170049
Title of the course
Digital Signal Processing - MUTUATO
Credits
6
Field
INF/01
Year of course
3
Pag. 11/22
Code
A5170037
Title of the course
Foundations of Geometry II
Credits
5
Field
MAT/03
Year of course
3
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Ferrario Davide luigi
Programme
Aim: the course is an introduction to some ideas and fundamental tools of modern geometry, such
as homotopy and homology theory, together with some examples of applications: classification of
surfaces, Morse theory and intersection theory.
Prerequisites: --.
Main Topics:
* Manifolds and atlas.
* Triangulation and triangulability, simplicial complexes.
* Graphs and trees. Euler-Poincaré characteristic.
* Homotopy, homotopy type and fundamental group.
* Classification of compact and connected topological surfaces.
* Singular homology and cohomology groups.
* Differential forms and De Rham cohomology.
* Integration on manifolds. De Rham Theorem.
* Introduction to Morse theory.
* Brouwer fixed point theorem.
* Winding number and topological degree.
* Borsuk-Ulam theorem.
* Introduction to intersection theory.
Bibliographic references:
J.M. Lee, Introduction to smooth manifolds, Graduate Texts in Mathematics 218, 2003.
J.M. Lee, Introduction to topological manifolds, Graduate Texts in Mathematics 202, 2000.
J.W. Milnor, Topology from the differentiable viewpoint, Princeton University Press, 1997.
I.M. Singer, J.A. Thorpe, Lecture notes on elementary topology and geometry, Springer Verlag,
1976.
Tests: written and oral test.
Code
A5170038
Title of the course
Foundations of Higher Analysis
Credits
5
Field
MAT/05
Year of course
3
Semester
n.d.
Assesment
method
Scritto; Voto finale
Lecturer
Cellina Arrigo
Programme
Aim: to introduce the main concepts of modern real analysis and of basic complex analysis.
Prerequisites: --.
Main Topics:
Real Analysis
* A review of Lebesgue integral.
Pag. 12/22
* Convex functions; polars. Normed spaces. Spaces of p-integrable functions and their
completeness. Holder's inequality.
* C(X) is complete; Continuous linear functionals; X' is complete. A linear functional is continuous
if and only if its null space is closed. Extension of functionals and Hahn Banach theorem. Baire's
theorem. The unit ball in infinite dimension is never compact. Weak topology and weak convergence.
Mazur's lemma; weak star topology. Alaoglu's Theorem. Hilbert spaces; best approximation and their
characterization. Complete orthonormal sets. Uniformly convex spaces.
Complex Analysis
* Cauchy Riemann equations; harmonic functions.
* Power series. Moebius maps.
* Line integrals.
Bibliographic references: bibliography will be given at the beginning of the course.
Tests: written test, consisting of four questions, three on real analysis and one on complex analysis.
Code
A5170039
Title of the course
Foundations of Higher Analysis
Credits
5
Field
MAT/05
Year of course
3
Semester
n.d.
Assesment
method
Scritto; Voto finale
Lecturer
Cellina Arrigo
Programme
Aim: to introduce the basic concepts of modern real analysis. To improve the knowledge of complex
analysis.
Prerequisites: --.
Main Topics:
Real Analysis
* Total boundedness and compactness. Ascoli Arzela' Theorem and its applications.
* Convolutions. Strong compactness in L1.
* An introduction to Sobolev spaces.
Complex Analysis
* Cauchy Theorem and Cauchy integral formula. Taylor and Laurent series. Liouville's theorem.
* The Fundamental Theorem of Algebra. The open mapping theorem.
Bibliographic references: H. Brezis, Analisi Funzionale - Teoria e Applicazioni, Liguori Editore, 1986.
Tests: written test, consisting of four questions, two on real analysis and two on complex analysis.
Pag. 13/22
Code
A5170070
Title of the course
Galois theory
Credits
5
Field
MAT/02
Year of course
3
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Weigel Thomas stefan
Programme
Aim: one of the central subjects in the theory of fields is the study of finite extensions, in particular
finite Galois extensions. In the times of E.Galois many mathematicians still worked on problems
formulated by the ancient greek mathematicians. One problem of this type was the trisection of an
arbitrary angle in the plane by ruler-compass constructions. From the Main theorem one concludes
easily that this is impossible.
Prerequisites: Algebra, I and II mod. Note: This course is independent of Instituzioni di Algebra I
mod.
Main Topics:
* The algebraic closure.
* Integrality and the integral closure.
* Normal extensions.
* Separable extensions.
* Galois extensions and the Galois group.
* Cyclotomic extensions.
* The Main theorem in Galois theory.
* Constructions with ruler and compass.
* Solutions of algebraic equations.
Testi: S.Bosch, Algebra, Springer, Unitext, 2003.
Exams: written and oral exams.
Code
A5170051
Title of the course
Image Processing (Advanced) - MUTUATO
Credits
6
Field
INF/01
Year of course
3
Pag. 14/22
Code
A5170048
Title of the course
Image Processing (Advanced) - MUTUATO
Credits
6
Field
INF/01
Year of course
3
Code
A5170043
Title of the course
Information Theory (Coding and Cryptography) - MUTUATO
Credits
6
Field
INF/01
Year of course
3
Code
A5170068
Title of the course
Institutions of Numerical Analysis
Credits
5
Field
MAT/08
Year of course
3
Semester
n.d.
Assesment
method
Orale; Voto finale
Lecturer
Bozzini Maria tugomira
Programme
Aim: to study from a theoretical and algorithmic point of view, some mathematical topics connected
with the image processing.
Prerequisites: --.
Main Topics:
* The theory of Plancharel.
* Analog and digital signals.
* Window transform.
* Theory and applications of JPEG.
* Applied numerical examples.
Bibliographic references: C.K. Chui, An introduction to wavelets, Academic Press, 1992.
Tests: oral test.
Pag. 15/22
Code
A5170065
Title of the course
Introduction to Modern Physics : II Mod.
Credits
3
Field
FIS/01
Year of course
3
Semester
n.d.
Assesment
method
; Voto finale
Lecturer
Sironi Giorgio
Programme
Aim: understanding the structure of matter.
Prerequisites: Maxwell Equations.
Main Topics:
* Review of Maxwell Equations.
* Crisis of the Classical Law of Physics: Special Relativity.
* Crisis of the Classical Law of Physics: Blackbody Radiation and Planck Law.
* Photoelectric effect, Compton Effect.
* De Broglie hypothesis, wave aspects of matter.
* Atomic hydrogen structure, Bohr atom model.
* Schroedinger equation: general properties.
* Atomic and Molecular structure: introduction. Electrical conductivity of solids.
* Nuclear Structure, Elementary Particles, Unification of Forces: introduction.
Bibliographic references: bibliography will be given at the beginning of the course.
Tests: rules will be explained at the beginning of the course.
Code
A5170060
Title of the course
Introduction to Modern Physics : I mod
Credits
7
Field
FIS/01
Year of course
3
Semester
n.d.
Assesment
method
; Voto finale
Lecturer
Sironi Giorgio
Programme
Aim: fundamental laws and interpretation of natural phenomena.
Prerequisites: notions of Mechanics and Thermodynamics.
Main Topics:
* Static Electric and Magnetic Fields: phenomena and definitions.
* Slowly variable Electric and Magnetic Fields: phenomena and definitions.
* Gauss Laws for Electric and Magnetic Fields.
* Ampere Law.
* Faraday Neuman Lenz Law: induction phenomena.
* Electric and Magnetic Fields in a medium.
* DC and AC circuits.
* Maxwell Equations: integral and differential form.
* Waves and electromagnetic waves.
* Propagation of electromagnetic waves.
Pag. 16/22
Bibliographic references: bibliography will be given at the beginning of the course.
Tests: rules will be explained at the beginning of the course.
Code
A5170036
Title of the course
Introductory geometry - I mod.
Credits
5
Field
MAT/03
Year of course
3
Semester
n.d.
Assesment
method
Scritto; Voto finale
Lecturer
Ghigi Alessandro callisto
Programme
Aim: The first part of the course will focus on smooth submanifolds of Euclidean spaces. The second
part will be centred on fundamental group and covering spaces.
Prerequisites: calculus in several variables, linear algebra, elementary topology and surface theory.
Main Topics:
* Implicit function theorem. Smooth submanifolds of Euclidean spaces. Tangent spaces and
tangent bundle.
* Immersions, submersions, local diffeomorphisms.
* Critical points and regular values. Sard theorem.
* Homotopy of maps. Homotopy type.
* Fundamental group of a topological space. The theorem of Seifert and van Kampen.
* Coverings (if time permits).
Bibliographic references:
E.Sernesi, Geometria 2, Bollati Boringhieri, 1994.
G.E.Bredon, Topology and Geometry, Springer, 1993.
V. Guillemin, A. Pollack, Differential Topology, Prentice Hall, 1974.
Tests: written and oral test.
Code
A5170042
Title of the course
Logical foundations of computer science (advanced course) - MUTUATO
Credits
6
Field
INF/01
Year of course
3
Pag. 17/22
Code
A5170041
Title of the course
Logical foundations of computer science (introductory course) - MUTUATO
Credits
6
Field
INF/01
Year of course
3
Code
517029
Title of the course
Mathematical Physics II
Credits
5
Field
MAT/07
Year of course
3
Semester
n.d.
Assesment
method
Orale; Voto finale
Lecturer
Noja Diego davide
Programme
Aim: the course introduces the main mathematical aspects of partial differential equations of
Mathematical Physics, Laplace, wave and heat equations. Applications are discussed, particularly in
Physics.
Prerequisites: --.
Main Topics:
* Introduction. Partial differential equations and definition of classical solution. Classical examples:
Laplace equation and static phenomena, heat and diffusion equation, wave equation. Other
examples from physics and applications.
* Linear first order equations and the method of characteristics. Initial and boundary conditions.
Well posed problems. Classification of constant coefficients second order linear equations.
* The wave equation on the line. Solution of the Cauchy problem and D'Alembert formula.
Influence and dependence domain and causality principle. Energy conservation.
* Heat equation on the line. Maximum principle. Energy methods, unicity and continuous
dependence of the solution on the initial data. Einstein description of the brownian motion and
connection with the diffusion equation.
* Wave and heat equation on a bounded interval. Dirichlet, Neumann and Robin boundary
conditions. Fourier series in Hilbert spaces. Bessel inequality and Parseval identity. The regular
Sturm-Liouville problem and its spectral theory.
* Introduction to the Laplace equation. First Green formula. Harmonic functions. Mean value
theorem. Maximum principle. Dirichlet principle and applications. Second Green formula and
representation of harmonic functions. Green functions and representation formulae for Dirichlet and
Neumann boundary value problems. The case of the halfspace and the ball. Poisson formula.
* Spherical means and Darboux equation. Kirchhoff formula for three dimensional wave equation.
Energy conservation. Huygens principle. Hadamard method and two dimensional wave equation .
Duhamel principle and the wave equation with external source.
* N-dimensional heat equation. Cauchy problem and solution formula. Schrödinger equation.
Harmonic oscillator and hydrogen atom.
* Heat and wave equation on n-dimensional bounded domains. Vibrating membranes and three
dimensional vibration of an elastic sphere.
* Eigenvalues, eigenvector and minima of the energy functional for the laplacian. Minimax
principle. Weyl estimates.
* Introduction to the theory of distributions. Test function and distributions. First operations with
distributions. Convolution. Schwartz class and tempered distributions. Fourier Transform of
Pag. 18/22
tempered distributions, Fundamental solutions of constant coefficient partial differential operators.
Application to mathematical physics and partial differential equations.
Bibliographic references:
W.A. Strauss, Partial Differential Equations. An introduction, John Wiley and Sons, 1992.
S. Salsa, Equazioni a Derivate Parziali, Springer Italia, 2004.
Tests: oral test.
Code
A5170059
Title of the course
Numerical Models
Credits
10
Field
MAT/08
Year of course
3
Semester
n.d.
Assesment
method
; Voto finale
Lecturer
Russo Alessandro
Programme
Aim: to study some mathematical models of the real world of interest for modern applications. The
analysis will be complemented by practical use of the Matlab toolboxes.
Prerequisites: the courses of the previous years.
Main Topics:
the text used is Introduction to Applied Mathematics, by Gilbert Strang (Wellesley-Cambridge Press,
1986). Chapters 2,3,4 and 8 will be covered.
Equilibrium Equations (Chapters 2 and 3)
* A Framework for the Applications.
* Constraints and Lagrange Multipliers.
* Structures in Equilibrium.
* One-dimensional Problems.
* Differential Equations of Equilibrium.
* Laplace's Equation and Potential Flow.
* Equilibrium of Fluids and Solids.
* Elements of Calculus of Variations.
Fourier Analysis (chapter 4)
* Spectral Approximation of functions defined on an interval.
* Discrete Fourier Transform and the FFT.
* Trigonometric Interpolation.
Optimization (chapter 8)
* Introduction to Linear Programming.
* The Simplex Method and Karmarkar's Method.
* Duality in Linear Programming.
* Saddle Points (Minimax) and Game Theory.
* Nonlinear Optimization.
Bibliographic references: G. Strang, Introduction to Applied Mathematics, Wellesley-Cambridge
Press, 1986.
Tests: rules will be explained at the beginning of the course.
Pag. 19/22
Code
A5170062
Title of the course
PROBABILITY II
Credits
5
Field
MAT/06
Year of course
3
Semester
n.d.
Assesment
method
Scritto e Orale; Voto finale
Lecturer
Bertacchi Daniela
Programme
Aim: conditional expectation. Discrete time stochastic processes: Markov chains and martingales.
Prerequisites: students are supposed to know concepts and tools usually developed in a Probability
Theory course, in particular the use of measure theory to modelize random data, the main
probabilistic models and convergence theorems.
Main Topics:
* Conditional expectation (with respect to a general sigma-algebra and with respect to sigmaalgebras generated by random variables).
* Markov chains with finite or countable state space.
* Discrete time martingales, stopping times and optional stopping theorem.
Bibliographic references:
W. Woess, Catene di Markov e teoria del potenziale nel discreto, UMI, 1996.
D. Williams, Probability with martingales, Cambridge University Press., 1991.
P. Baldi, Calcolo delle probabilità e statistica, McGraw-Hill., 1998.
P. Billingsley, Probability and measure, Wiley & Sons., 1995.
R.P. Laha, V.K. Rohatgi, Probability theory, Wiley & Sons, 1979.
Tests: written and oral test.
Code
A5170047
Title of the course
Processing (Fundamentals) - MUTUATO
Credits
6
Field
INF/01
Year of course
3
Pag. 20/22
Code
517028
Title of the course
RELATIVITY
Credits
5
Field
MAT/07
Year of course
3
Semester
n.d.
Assesment
method
Scritto; Voto finale
Lecturer
Magri Franco
Programme
Aim: the purpose is to give an introduction to Einstein's Relativity, from the viewpoint of the geometry
of spacetime.
Prerequisites: no preliminary acquaintance with the subject is required. Nevertheless the
understanding of General Relativity is made easier by the knowledge of the basic notions of the
intrinsic geometry of surfaces, as presented in the course of Differential Geometry.
Main Topics:
* The Maxwell theory and the crisis of Classical Physics. The axioms of Special Relativity. The Kcalculus, and the discovery of the geometry of Minkowski's space. Lorentz transformations and
some of the relativistic paradoxes.
* The intrinsic formulation of relativistic dynamics. Four vectors and the equivalence of mass and
energy.
* The intrinsic formulation of Maxwell theory. Maxwell and Faraday 2-forms and Maxwell
equations written with forms.
* A brief introduction to the geometry of curved spacetimes. Vector fields and tensor fields.
Geodesic and curvature.
* The Einstein's theory of gravitation. The principle of equivalence and its implications. The
Einstein's equations. The Schwarzschild solution and its maximal extension. The precession of the
perihelion of Mercury and the deflection of light.
Bibliographic references:
G.F.R. Ellis, R.M. Williams, Flat and Curved spacetimes, Oxford University Press, 2000.
S. Carroll, Spacetime and Geometry: an introduction to General Relativity, Addison Wesley, 2004.
Tests: oral test (exercises are done during the course, and they are an integral part of the course).
Code
A5170076
Title of the course
Algorithms and programming - MUTUATO
Credits
8
Field
INF/01
Year of course
n.d.
Pag. 21/22
Code
A5170075
Title of the course
Introduction to Data Bases - MUTUATO
Credits
7
Field
INF/01
Year of course
n.d.
Pag. 22/22